Production Analysis: Econometric Issues by Ronald Yacat

Number Twenty-Three,

Volume X I II, 1986

PRODUCTION ANALYSIS: ECONOMETRIC ISSUES

Via. Cynthia 5. Bantilan

This paper addressessome econometric issuesthat arise in production analysis. The first section presents the basic analytical framework in analyzing production decisions. The discussion serves as a background for the three subsequent sections which deal with the following issues: model selection, the use of panel data, and bias in nonrandom sampling.

Methodological Foundation

There are essentially two approaches in econometric productivity analysis, namely, the primal and the dual approach. The first approach is based on the idea that, given a transformation or production function and the assumption of profit maximization or cost minimization, factor demand and output supply may be derived based on the necessary conditions of optimization. While this approach may be easily worked out from simple production functions (e.g., factor demandsare straightforwardly obtained from the Cobb-Douglas production function), it may not be workable when the production technology is complex. The second approach provides an alternative in obtaining relationships describing producer behavior. By appealing to duality theory and the concept of cost and profit functions, it enables one to derive a system of input demand and output supply quite readily while preserving the complexity of the structure of the producer decisionmaking Assistant Professor,Departmentof Economics, College of Development EconomicsandManagement, Universityof the Philippinesat LosBa_.os. 77

JOURNAL

OF PHILIPPINE

DEVELOPMENT

process. In the context of production studies, the duality theory provides the theoretical background that makes it possible to recover information about the production structure from a dual profit or cost structure. It theorizes that optimal input and output levels implied by the technical transformation function and the necessary conditions are equivalently obtained by the optimization of a dual profit or cost function. It ensures that when certain restrictions hold for the optimized dual function, they also hold for the transformation function. The general development of the application of duality theory and the concept of profit and cost functions is given in Section II of the overview paper. An application of these developments is illustrated as follows. A system model may be constructed for a farmer producing one output, say rice, by applying four variable factors of production, say labor, fertilizer, animal power and tractors..Consider that this farmer produces under perfect competition in both input and product markets. Assume that

g (Y, x 1,x2, x 3 , x 4 s) =o is the technical transformation function that represents the farmer's production technology, where Y represents Output, X i represents the ith input, i = 1,2,3,4 and S is a vector of variables representing the structure of the farm as may be measured by size of farm, proportion of area planted with high-yielding varieties, irrigation investment, past investment in researchand extension, and degree of rural electrification. Diewert (1974) proposed that, if the transformation function obeys certain regularity conditions, it is possible to obtain the maximum profits in production as. _r='/r

(P_ Wl,

W2, W3, W4, S,)

where P is the price of the product and Wi, i = 1,2,3,4 are observed factor prices. This profit function can be viewed as a solution to the following constrained maximization problem 7r(P_ W,S) =

g(v,

max

I(PY-

W3X3 - W4X4"

. . x 4,s) = o).

In this formulation it is assumed that, given an exogenously determined vector of input and output prices, farmers will choose the level of out-

BANTILAN:

PRODUCTION

ANALYSIS

put and factor combination that maximizes profits subject to the constraint presented by the production technology. This is the basis of the concept of duality between maximized profits and the technical transformation fun ction. Given the maximized profits function, the application of the Shephard-Hotelling lemma (Varian 1978) produces a system of equations representing the output supply and input demand functions. According to the lemma, optimal input and output levels are obtained by differentiation of the profits function with respect to output and input prices, i.e., Y= aTr/aP X i = c37r] aWl, i= 1,2,3,4. Note that the maximized profit, _r,are expressed as afunction of prices and structure variables. The expression does not include endogenous or choice variables such as output quantities and variable factors of production. These are eliminated from the expression by substitution of "first-order conditions." Consequently, the derived input demand and output supply equations are expressed as a function of prices and structure variables. The methodology described above has the following useful consequences. First, the procedure takes into account the nature of farmers' production decisions whereby choices of individual factor input and output levels are interdependent. The interrelationship among factor and product markets is captured in the derived system so that it provides a basis for a formal analysis of the simultaneous effects of policy changeson product supply and factor demand decisions. Second, the duality principle has allowed us to move from a transformation function which is a function of endogenous variables (quantities) to a profit function which is a function of exogenous variables (prices, fixed factors). As a consequence, simultaneity problems are avoided. Furthermore, multicollinearity problems are reduced since there is less c6,variance in prices than in quantities. Lastly, the functional forms for the Systems can be linear and economical in parameters and can still be "flexible" in the sense that they do not impose some restrictions on the "curvature" of the primal functions. Parameter estimates for the system of equations representing output supply and demand for various factors may be obtained by applying iterative versions of seemingly unrelated regressions (Zellner 1962;

JOURNAL

OF PHILIPPINE

DEVELOPMENT

Gallant 1975). Each equation in the system looks like an individua regression but the variances of parameters are reduced by simbltaneously estimating the system of equations so.that error interdependence is taken into account. The basic,assumption in this procedure is related to serial independence and contemporaneous correlation of the residual terms. If etj is the residual term corresponding to the observation t in the regression], and et" j" is the same for observation t' in equation j" , the following implication will be true with respect to the variance-covariancematrix E (etj, et,j" )

=0ift ajj,

= t" if t = t"

This method is an extension of the generalized least squares procedure. It is preferred because it permits convenient imposition of across equation restrictions. Examples of studies that used the above methodology are Lau and Yotopoulos (19"/2), Yotopoulos, Lau and Lin (1976), Sidhu and Baanante (19"/9, 1982), Quizon (1980), and Kalirajan and Flinn (1983).

The Problem of Model Selection One issuethat immediately comes up in the empiricalapplication of duality theory is the choice of functional form for the statistical model of the unknown cost or profit functions. Statistically, we seek a form which gives closeapproximations to the parametersof interest. The problem is to determine which mathematical formulation can provide the flexibility required on the parameters of the function. To this end, the use of functional forms that are flexible, meaning that the parameters can take arbitrary valuessothat it doesnot necessarily impose restrictions on the curvature of the production technology, is widespread. Flexibility is based on approximation theory. Two methods of approximation are usually used: Taylor series approximation and Fourier series approximation. In fact, the functional forms_are referred to as either locally flexible (Taylor series) or globally flexible (Fourier series)accordingto the method of approximation used.

BANTILAN:

PRODUCTION

The notion of ted from Diewert's local approximation, assatisfaction of the

ANALYSIS

"local flexibility" in production analysis origina(1974) suggestion of the use of a second-order say g, to the true function g* at some price P* following conditions: g (P*)= g* (P*) _g(P) I

a2 g(p)

= ag* (P)

= a2 g.(p}

These conditions can be understood in two ways. In the first place, it can be taken to mean that the first-and second-order derivatives of the actual function, g* (P*), are equal to the derivatives of the approximating one, g (P*),, at the point P*, for example, the point of profit maximization. The alternative criterion is that, in addition to the equality of derivatives at point P*, the deviation between g* (P*) in a defined neighborhood about the point P* consists of only terms of the third or higher order. This implies that any second-order Taylor series expansion about point P* is a second-order approximation, as the deviation is bounded by the remainder term. In general, functions that can be considered second-order Taylor series about the point of profit maximization are in fact second order approximations to the underlying flexible aggregator function at that point by either of the two meanings of the term given above (Lau 1974, pp. 183-184; Fuss et al. 1978, pp. 233-234). It is important to note that Diewert's definition of flexibility does not require a quadratic form or a Taylor approximation - only a twice differentiable expression. The flexible forms commonly used for the dual relationships are themselves nonlinear but have linear derivatives. Forms with linear derivatives are not generally "globally convex," i.e., convex at all possible data points, but are convex over certain ranges. In practice, however, the most commonly used locally flexible forms are essentially quadratic forms and take a Taylor series interpretation. A description of selected functional forms is given in the appendix. For example, take the translog cost function i.e. InC=f3 o + Zfii InW i + (112) _i Ei/3ij In Wi In Wj.

JOURNALOF PHILIPPINEDEVELOPMENT

This may be viewed as a second-order Taylor Series approximation of some unknown cost function by comparing the above form with the, following. , InC = lnC

InWo

+ _i

alnC

(InWi-lnWo)

alnWi

InWo

+ (1/2)zizj

M (Inwj-/nWjo). aln Wi aln Wj

In Wo

The standard practice is to regress In of cost on In of prices; regression coefficients are then interpreted as the coefficients in the Taylor series, i.e., InC = _o

alnc a/nwi a2 Inc .=

f3ij

aln Wi _lnWj the

A critique of the use of the Taylor construction of flexible functional

series expansion as a basis for forms is made by Gallant

(1981). The argument is on two levels. In the first place, the Taylor expansion is only an approximation over a nonspecified (unknown and unknowable) region -- its properties as one moves away from the point of approximation cannot be ascertained with any degree of confidence. In the second place, the econometric properties of the Taylor

series are weak (Gallant

1981, p. 212).

Taylor's theorem fails rather miserably as a.means of understanding the statistical behavior of parameter estimates and test statistics. If one insists on using Taylor's theorem as a means of understanding statistical behavior one is led into an algebraic morass.The reason for this failure is that the statistical regressionmethods essentially expand the true function in a (general) Fourier series - not in a Taylor series. Due to this fact, Fourier series permit a natural transition from demand theory to statistical theory. The key fact which permits this transition is that the classical Fourier sine/cosine series expansion approximates not only the function to within arbitrary accuracy.., but also its first derivatives.

BANTILAN:

PRODUCTION

ANALYSIS

Other authors criticize the practice (White 1980 and Barnett 1976, among others) by taking issue on the actual behavior of a Taylor series approximation in a regression setting. On the one hand, Taylor series approximants apply only locally about a point or about its neighborhood. The approximating error from Taylor series approaches zero as the point of approximation is reached but can become quite large away from the point of approximation. On the other hand, regression methods attempt to make errors in approximation small over the range of data, that is, the function is usually not fitted about a specific point, but over a domain or a constructed sample mean. This may entail that the function in fact is not a second-order approximation at any particular point. The relationship between the real and fitted functions, therefore, will contain an element of uncertainty as to the interpretation of some comparative static results. So, while the approximation method applies locally about a point, the estimation procedure applies globally over the range of data. The implication is that estimates obtained in this manner do not consistently approximate the parameters of the true function (unless the latter is of the approximating form). Elasticities obtained are as a consequence also inconsistent. If the true function is actually of the translog form, for example, then there is no problem; and the procedure is straightforward curve fitting of the cost function. If not, then one interprets the flexible functional form as an approximating function -- the coefficients of which are interpreted as the derivatives in Taylor series expansion. As pointed out by Gallant, the empirical function is constructed to be an approximation about a certain point of the true aggregator function, but it may not maintain that desirable property as we move away from the original point of approximation. For other possible sets of positive finite price vectors outside a narrow range about the original, such crucial properties as convexity and monotonicity may fail. Since modelling implies extrapolating outside this narrow range, the possible loss of approximating characteristics in the empirical analyses may weaken the quantitative and perhaps even the qualitative results obtained. 1 Under certain conditions, the fitted function 1. For a discussionof thesepoints, seeLau (1978), wherehediscusses the problemsof convexity and monotonicity for the threefunctionsforms usedin this study and developsteststhat canbeusedex post.The article by DouglasCavesand LauritsChristensen(1980) citesthe lack of theoreticaldiscussionof globalpropertiesof the flexible functional forms, sincethe literature hassofar only referredto

JOURNAL

OF PHILIPPINE

DEVELOPMENT

may in fact reject a test for the symmetry constraints, even though the constraint may be true for the actual function (Fuss, McFadden,. and Mundlak 1978, pp. 233-234). 2 These observations led to an alternative approach to the approximation problem: the Fourier form (Gallant 1981). It uses the Sobblev norm as the appropriate measure of approximating error. The Sobolev measure of the distance of an approximating function,g (x), from the true g* (x) is l19*-g llm, P, w = (Z f I DUg *lul* <m

DUg lPdw(x)

) (lIP), I <P<

where m denotes the largest order derivative of g which is of interest, â&#x20AC;˘ Du denotes partial differentiation, w(x) is a distribution function, and u is the region of approximation. This means that, ifg approximates g* poorly or poorly approximates any of its derivatives to the order m, then the Sobolev norm will assigna large value to the approximation error (Gallant 1984). The relevance of the use.of the Fourier form in production analysis is the interestâ&#x20AC;˘.in approximating not only the unknown function but the first .and second derivatives as well. For example,.an estimate of the measureof the following crosselasticity of substitution a2C awi

awj

0=1+ ac

aWi

awj

empiricaltests of the variousforms now currently beingused(see,for example, Ernst Berndt and MohammedKhaled 1979). Cavesand Christensencarry out a discussionof thesepropertiesfor the translogand linear functionsas the regions over which the regularity conditionshold in the two casesaredifferent. The discussion is carried out with respectto the variation in price elasticitiesand substitution, and givea graphicalexpositionof the comparisonof theseforms in the two andthreecommodityhometheticand nonbomotheticcases. 2. If the function is in fact fitted abouta singleon the actualfunction, and not at an estimatedmean, then the approximatingfunctions shouldconfirm the symmetry condition and any other constraintswhich the actualfunction fulfills. SeeLau (1978,pp. 409-I I, 418-20).

BANTI LAN: PRODUCTION

ANALYSIS

(where C represents a cost function C = C(W) and W is a vector of priceS) is of economic interest. One can see from this expression, that it is not enough that the COSt function, C, approximate the true cost function closely; the first and second derivatives of the approximating function mustalso-appt'oximate the first-and Second derivatives of the true cost function closely. The Sobolev-measure of approximation error takes into account also.the need to approximat_e derivatives. The Fourier series expansion has the global propei'ty Of approximating a function throughout its domain..This lends itself naturally to the usual practice of running, a regression to estimate parameters because now both the method of approximation, and estimation procedure are global. Gallant (1981) shows l_hat estimation based on a Fourier series produces consistent estimates of the parameters of interest. There have been questions about how good are the approximations that these .flexible forms provide.. Guilkey and Lovell (1980) and Gullkey, I_ovell andSickles (1982) have shown that, if a tranSlog approximation has elasticities of substitution that depart fi-om unity, the quality of the form deteriorates markedly. Guilkey and associates explain that there is a tradeoff between flexibility and global behavior: â&#x20AC;˘When selecting a functional form for use in empirical work, one is confronted by a choice between forms that exhibit global behavior and those that possessflexibility. The use .of flexible functional forms based on Taylor's expansions implies the possibility of not satisfying, the regularity conditions globally unless the true function is identical to the chosen representation. If one â&#x20AC;˘ wants models that are flexible, there is the risk ofhaving forms that do not obey those conditions over all Observations. Onegood example of this tradeoff can be demonstratedby the translog specification. A sufficient condition for global convexity is to have all second-order coefficients equal zero; however, this leads to an inflexible Cobb-Douglas functional form. With respect to the use.of Fourier forms, substantial tradeoffs between bias and instability have been indicated. Chalfant (1984) shows that the Fourier form, with desirable properties concerning bias, features much greater oscillation in estimated elasticities t.han does the generalized Box-Cox (which nests the generalized leontief and generalized square-root quadratic forms as special cases and the translog as a limiting case). Further research is called for to adjust estimation procedures and model specification to improve the attainable levels of unbiasedness and stability,

â&#x20AC;˘ 86

JOURNAL

OF PHILIPPINE

DEVELOPMENT

On the Use of Panel Data A data base that uses time-series and cross-sectioninformation is oftentimes useful in production analysis. They are important tO analysts because they contain information that is necessarytodeal with both intertemporal dynamics and individual effects Of entities being investigated. In addition, time-series information alone or cross-section information alone may not provide enough variation or is insufficient to provide enough degrees of freedom for fitting especially in cases where more complicated flexible functional forms are involved. Pooled time-series and cross-sectioninformation (panel data) may thus make econometric estimation feasible. The most common approach in the analysis of panel data is the use of the covariance or dummy variable model. The basic approach is to identify cross-sectionsby dummy variables and then to apply the generalized least squares estimation method (Zellner 1962) to obtain the parameter estimates.This method is computationally appeal_ ing and yields asymptotically unbiased and consistent estimators (Wallace and Hussain 1969). In principle, the weaknessof covariance, approach lies in the fact that inter-cross-section variation is ignored, i leading to inefficient estimation when this variation is known (Fuss, I 1977). For a complete review,of the current statistical methodology l on dealing with pooled cross-secti0nand time-series data, see Dielman

(1983). One item of serious concern in empirical work is the problem of errors in variables.Consider the equation, Yit = i_i + ([JXit- {Jvit) + nit = I_i + {JXit + eit . where

(1)

#i 's are unobserved individual effects; vit is an i.i.d, measurementerror;

and

nit are the standard best case disturbance term --- (0, as i).

Parameter estimates obtained from using OLS estimation of the above equation which involves erroneously measured right-hand variables will be biased for two reasons: (1).because of the correlation of the )(it with â&#x20AC;˘ the left-out variable effects; and (2) becauseof the negative correlation between the observed Xit and the composite disturbance term (nit-

vit).

BANTI LAN : PRODUCTION

ANALYSIS

Griliches and Hausman (1984) proposed a clever way of identifying the magnitude of the "true" parameter in a panel data context. The key idea is that there are different ways of transforming the data to eliminate the first source of bias; and that different transformations imply different and deducible changesfor the magnitudes for the second type of bias. Such changes can be used in identifying the magnitude of the true parameters. They propose the use of difference models of the form. dy

d X fl-

dvfl

= d2y=d2X(3-d2v_+d2n d my

= d taxI3 -

+ d n = d X fl + de =

(2)

d 2 Xfl + d 2 e

d m v8 + dmn = d m X_ + dm e

where d nyt

Yit

Yit-n

to obtain implicit estimates of the bias..For panel data with a time series of length T, one can construct 7-/2 difference models from which there are 7-/2 independent consistent estimates. The strategy is to take advantage of these alternative consistent estimates, they are compared to obtain estimates of bias. The general procedure suggested is given in three steps. First, estimateequation (1) by generalized least squares (variance compo-nents) and by within estimation. Do a test for equality of the estimates using a Hausman (1978) or Hausman-Taylor (1981) type test.Second, if you reject the hypothesis regarding the equality of estimates, then calculate some differenced estimates by OLS. If they differ significantly, errors in measurement may well be present.' A joint test of all the differenced estimates can be made by using GLS on the system of equations in (2). Lastly, estimate the equations in (2) using the instrumental variable technique. In this respect, the X's provide the instrumental variables for each equation where all X's not involved in the difference are used as instruments. Then do a specification test of the no correlation assumption in the errors in measurement. If the different instruments of fl do not differ significantly, then the magnitude of the true parameter estimate is obtained. If they do differ significantly, one of the following is called for: the specification of specific correlated errors in measurement process, the useof outside instruments, or the respecification of the original model.

JOURNALOFPHILIPPINEDEVELOPMENT

Bias from Non-Random

Sampling

Randomization in the selection of sample units ensures two things. It eliminates subjective selection bias and provides a basis for statistical inference. This process, however, is oftentimes not realiZed in the implementation of practical survey work. In some cases, sampling units. are Simply selected for convenience without randomization. In other cases, as in the case of complex surveys, it is economical to select existing natural groupings of observations. These are characterized by relative homogeneities within the groups that negate the assumption of independence of sample elements. In a regression setting, the situation is equivalent to the failure of satisfying the assumption of independence betweeen observationsand the bias in parameter estimates that followS. Consider a View of the problem taken by Kish .and Frankel (1974): (i) Consider a finite population of sizeN. Associated with each of these elementsis a vector of h + i Xli,

values YI' X/i,

S2it

"",

)(hi.; P(fYi'

X2i , ... , )(hi) ). (ii) Our parameters are numbers Bj such that _i N

(Yi " _ _ Xj_ = is minimum subject to Zy (Yi" _jh _j)_._ ---0, (iii) Given a sample of N vectors from the population of N vectorS,our desire is to estimate the parameter _.. The regressionmodel stated in (ii)' does not in practice correspond exactly (or'.even closely) to the complex, relationship among the actual population'of'vectors. The error term measures(usually in a least-squares sense)the extent to which the model departs from the actual complex relations among the population of vectors. Statistical theory of regression assumesa basic,structure of relationships. Letting X I = (Xli. , .... )(hi) and _ = (_1, "", _h), it. usesthe model YI =X_ +e i and then makesseveralstrong assumptions: A. linearity B. homosce.dasticity C. independence between observations D. normality for the ei. Assumptions (A)., (B); and (D) concern the basic.structure of the universeof the model, whereas(C) involves independent selections from it. This (or a similar) well-specified model yields desirable results; the standard least-squares b are minimum variance, linear, unbiased, normal, etc. Literature and textbooks are written about this pretty model; this is what researchersfind but they find very little to reconcile this model with the real population they are investigating.

BANTI LAN: PRODUCTION

ANALYSIS

Specifically, assumption (C) fails to hold when correlation between observations is induced by the sampling scheme or sampling. implementation. Recognizing the problem as such, it looks like the correction for bias may be implemented by standard econometric methodology. Directions in tackling the problem are indicated by the studies of Holt, Smith and Winter (1980) and Kish and Frankel (1974). In Holt, Smith and Winter, they assume Y = X_ + e ,'l Ere/X) = 0; B = (X'X) "I (X'Y) = _8 + 0 (]/_/N)inafinitepopulationofsizeN, where/_ is the corresponding parameter on the infinite population from which N came. If (Tri, i = 1, . . .,..N) denote the inclusion probabilities, they find that an element of

Y lVi Xij Xik of X" X and Z;N XI] Yi

of X" Y have probability weighted estimates, F,XiiXik /Tri and F,Xij Yi/ 7rpHence a probability weighted estimator of fl is b* = (X'D'X') where D=

'-i

fX'D -1 y)

1 7r2

In the caseof Kish and Frankel, the_/ ,,16

_.,jsXjYj/ 7rj Z,jsXy2 / _rj

where _/ is defined as selection probabilities instead of inclusion probabilities. The direction that is suggestedby the above studies is to correct the bias analytically by incorporating known or assumed probabilities of selection of samplingunits. Econometrically, this may be implemented â&#x20AC;˘in two ways: (1) reparameterize the model to integrate the available information; and (2) incorporate the information into the error structure of the model. For example, for the model Y = X_ + e, use the fact that, for every positive semidefinite matrix D, there exists a matrix P such that (p'p)-l=D. Thus, one can construct a reparameterization matrix P

JOURNAL

OF PHILIPPINE

DEVELOPMENT

which integrates information about the sampling scheme, e.g., •known or assumed selection probabilities. In the problem considered by Holt et el. (1980) and Kish and Frankel (i974), the matrix P may be constructed as 1 0

_/_I p ._-

_/_2 1

v'_n

so that the model may be reparameterized as

Y*= X*_ + e* where

Y* = PY X* = PX

In effect, the observation matrix is weighted by the inclusion probabilities. With the reparameterized model, GLS estimation would give

O--(x'Dx)-'

(x" o Y).

•The author notes that, in the studies cited, no explicit derivations or explanations were given. The results, however, are consistent with such reparameterization. APPENDIX Themostcommonlyusedflexiblefunctionalformsusedin studiesto dateare the transcendental logarithmic • function (translog),the normalizedhomogenous quadraticfunction,the generalizedLeontiefor linearfunction,the Generalized Box-Cox,andthe Fourierflexibleform.

BANTILAN:PRODUCTIONANALYSIS A.

The Tmn$/ogFunction

This function was first introduced by Christensen. )orgenson and tau (1973) -and developed further by Lau (1978) "1.The unrestricted profit function form is written as InTr= _,i bi InPi + El _j bij InPi /nPj Since we are concernedwith the short-run profit maximizing behaviorof farmers, we want to use the restricted form. In addition, certain restrictions have to be imposed to attain the property of linear homogeneity in prices. It has been shown that the translog function is linearly homogeneous if and only if (Diewert 1974, p. 139): _lb i

= 1, and

_,ibij = 0 (and by extension _l_ij

= O)

The restricted translog function is then given by InTr -- _,ibilnt_ _,iZhbl,

+ _,i_/bl/In lnPiZh

Piln_

+ T_ibittlnP

+ i

It turns out that this function is globallyconvex only if all bij = O, that is, the func. tion degeneratesto a Cobb-Douglas form with unitary elasticity of substitution and constant returns to scale(Lau 1974, pp. 182-83; Lau 1978, p. 240; Diewert 1974, p. 115). In most cases,however, it is possible to obtain a set of constraints on the bij coefficients which leavesthe function locally convex. When deriving the input demand and output supply functions in the translog case,the functions are expressedin value-shareform, since

alnl alnpi =ximil

=si

The sum of the value shares comes to zero since both inputs and outputs are included so only (rJ4) equations are linearly independent. Using the homogeneity constraint we therefore have:

a/n_/alnP i = si = a, _'t_bi/_ _jn-]bij(/nPi-/nP/) Z_ +. bitt

1. In some instances,all the three functional forms given here are written with a constant term. This is, strictly, not correct, since the functions then are no longer linearly homogeneouswith respect to prices - some exogenousincome is assumed.

JOURNALOF PHILIPPINEDEVELOPMENT

The coefficients bij do not have any explicit economic interpretation l but allow for easycomputation of the price elasticites we want. In generalwe have=:

. ij =

mix)

In the translog case, the cross price elasticities (CPE) and own price elasticities (OPE) therefore become: n ij = bij/Si

+ Sj

n ij = bii/ Si + Si-

(CPE) 1

(OPE)

Becauseof the fact that value sharesmust sum to zero, S must be estimated residually. B.

The Normalized Homogenous Quadratic Function

The normalized homogeneous quadratic function has the homogeneity constraint imposed by normalizing profits on the nth commodity. This commodity can be either an input or output - the normalization does not affect the optimum levels of inputs and outputs derived via Shephard's lemma. The function in its unrestricted form reads: fr/ Ptl = _n_, bt(Pi/ Pn) +(1/2" ) _v_-/,_-I

bij(Pi/

Pn ) (pj/ Pn)

If we use the convention that the tilde "_-_' abovea symbol indicates normalization on commodity n, the normalized homogenous quadratic function in its restricted form is

X n-1 bitt p P_ The necessary and sufficient condition for global convexity is that the symmetric matrix is positive semidefinite (Lau ] 974, p. 1 82). This property can be tested for using Lau's approach (Lau 1978). Becauseof the normalization, this form also has only (n--l) independent supply and demand equations of the form X i = a_l_

-- b i + _n-ibij_

j 4- _'hbikZk

4- bltt

2. Note that sincewe in fact have random (estimated) variables,both numerator and denominator, we can strictly speakingopt not to use methods basedon linear estimation techniquesfor testing the significanceof the elasticities(t.statistics,F-statistics). _;

B_.T, LA.:P.ooucT,oN ANALYS,S

The nth equation must becomputed residually, setting so that it becomes

x. = -

X n = _" -

_, n-, Xi _i' i

=";'oij55

The elasticities for the normalized homogenousquadratic function are thus nij = bij (_j / X i ) i =/=j,

i =/=n

(CPE)

0#=bii _lX; i _ n

(OPE)

The elasticitiesfor the nth equation are estimated residually, using the homogeneity constraint T.,jnij = ]_inl/ = O:

nnn = __ _, n-1 j nn) C.

(OPE)

The (Truncated) Linear/Leontief Function The General Linear/Leontief

_= T',_iPI(I/2)=

Function is written in the following form: 3

+T'iZjbijOi(1/2)

Pj(1/2)

The General Linear/Leontief function poses a problem, as it is a Taylor seriesexpansion in the square root of prices, and hence doesnot exhibit the linear homogeneity done with the homogeneousquadratiC.Here we will usethe truncated version instead, by setting all b=0, as this function still exhibits the second order approximation conditions (Diewert 1971, 1973). The derived demand and supply equationsare then:

Xl= _r /aPl _ bU+Z,jbij (Pj/Pi) (I/2)+ T.kbikZk+bitt by imposingthe sideconditionsfor the restricted profit functionas for the other two functions. The elasticitiesin this casebecome:

3. See Lau (1974, p. 184). Diewert (1973, pp. 295-300) between the gent: ralized linear, which he writes in a form similar to the one usedhere,and the generalized Leontief, which he writes as/r = _i_"bi 'P'Y2P:Â˝'I In this 1971 article, DieJ wert representsboth of them in the trunca_e_verslon (Diewert 1971, pp. 497505). Lau gives the more generalversion, as do Fuss, McFadden, and Mundlak (1978, p. 238) andit is only this generalizedversionwhich isa properTaylor Series form.

JOURNAL OF PHILIPPINE DEVELOPMENT

nij

(bii/2Xi)

(_/Pi)'/2

j=/=i

(CPE)

nij = - _j (bq/2Xt) (P//Pi) _ j=/=i (OPE) because of the homogeneity constraint, N is estimated residually, as no independent means of calculating it are available. D.

The Generalized Box--Cox Of the class of second-order

polynomial

flexible

eralized Box-Cox is the most general to date. and generalized square root quadratic forms appears as a limiting case. In various forms it Kiefer (1976), and Berndt and Khaled (1979). Berndt and Khaled. The expression for total cost

functional

forms, the gen-

It includes the generalized Leontief as special cases, and the translog has been applied by Denny (1974), The presentation here is taken from

C = [1 + X G(P)] I lXyB(Y,P) with

G(P) = %

+ T.i=inoliP i ( ) + (i/2)Zi=I n _'

P =vector

=1 n

_j P/ (X)Pj(X),

of input prices,

and B(Y,P) =/_ +(e/2) In Y+ _:i=1n _i InPi, Pi (X) = Pi ?_/2" ,!

X/2 with

the restriction

the following

that the cost function

restrictions

is linearly homogenous

in input prices,

are introduced:

_-'i= ln _i = 1

Xe_O,

T.j = 1 n3' ij = (;k/2) ei and _,j=l

n _i = O.

The cost function

then reduces to

C = [ (2/X) _;/.= 1

_'j=ln'Yij

x/2

x/2] 1/x_B (Y,P).

The term B (Y, P) involves interactions between prices and output, thereby producing a nonhomothetic technology. Homotheticity requires that each 6i be zero as in the translog case, and homogeneity, of degree (1/_ a_ih follows when O equals zero.

8ANTILAN: PRODUCTION ANALYSIS

To introduce technical change into the generalized Box-Cox cost function, Berndt and Khaled multiply by the term eT (?,

where T (t, P) = _ * ZI=

ri inPi

The analogy to the translog specification bias and so on.

is evident; t i represents technical change

Special cases of the generalized Box-Cox are obtained by fixing the BoxCox parameter _. When ;k equals one, the Berndt and Khaled specification reduces to the generalized Leontief. The generalized square root quadratic is obtained by setting _k equal to two. Finally, the translog is a limiting case as :k approaches zero. Expressions for factor shares in the generalized Box-Cox cost function are produced by differentiation according to Shephard's lemma.. The share of factor i is given by

Si=

Pi X/2

[Y'j=ln_ij

[y:i=ln Substitution differentiating

pj )'/21

__,j=lnijP

elasticities

+ 8i

i h/2/_.

given by Berndt

InY

+ r i t.

X/2] and Khaled

all = 1 -- 2_+"7ij (PIP/)

X/2

C-X +_

F j (%

ss/ Sj

Fourier

Flexible

I _'<-;(l';") Si ' iw

Setting _t equal to zero, this reduces to the translog su bstitu tion. E.

(1979) can be obtained by

once more and rescaling:

expressions for elasticities

Form

The logarithmic version of the Fourier (1982) takes a cost function of the form.

g#(Xle)= %+b'X+ (ll2)X'CX+ â&#x20AC;˘X , [#jc_COS

sk_ x)-

flexible

form

introduced

Z;e= 1{ "oe+ X ,

Vj_sin(j

sk_

]cj= X)I}

in Gallant

JOURNAL

OF PHILIPPINE

DEVELOPMENT

P1 whereX =

P2 Y

C=-XS2 Kt

X 8 b C A

= = = = =

_;<x= 1 #oO_k _

" _

number of parameters,possibly a function of the sample size determined by choiceA and J vector of exogenousvariable vector of parameters vector of own elasticities matrix of crosspriceelasticities number of multi-indices

Prior to estimation,it is necessaryto rescale4 the data becausethe Fourier function isperiodic,sodata mustfall within (0, 2 _r). Differentiation of the logarithmic Fourier flexible form producesshareequationsof the form

v qkt_(x/o_ b - xs _A +2Z/=

•( _o_Xs k(_x J[l_jotsin(jXsko_X)

+vj_cos (JXs_x) 1_ks where _7 denotes the gradient vector representing factor shares formed by dif•ferentiating with respect to the logged input prices. With the cost function and n- 1 share equations formed, parameters may be estimated with the seemingly unrelated regressionstechnique. 4. Recalling is to be accomplished by the following procedure: First, from each member of the logged seriesof exogenousvariables, subtract the minimum of 5 that seriesand then add some _ say 10" . Next rescaleany covariatesuchasoutput by a scalar

max. rescatedprice + EE

½= InYrnax - InYmJn +_" Finally, all data are rescaledby 6 maximum rescaledprice = + E_

BANTILAN:PRODUCTIONANALYSIS

REFERENCES

Barnett, William A. "Maximum Likelihood and Iterated Aitken Estimation of Non-linearSystemsof Equations." Journal of the American Statistical Association 7 (1976): 354-60. Berndt, Ernst, and Mohammed Khaled. "Parametric ProductivityMeasurementand Choice Among Flexible Functional Forms." Journal of Political Economy 87 (1979): 1220-40. Caves, Douglas,and Laurits R. Christensen. "Global Propertiesof Flexible Functional Forms." American Economic Review 70 (1980): 422.32. Chalfant, james A. "Comparisonof Alternative FunctionalFormswith Application to Agricultural Input Data." American Journal of Agricultural Economics. 66 (1984): 216-20. Christensen,Laurits R.; Dale W. Jorgensen;and Laurence,J. Lau."Transcendental Logarithmic Production Frontiers." Review of Economics and Statistics 65 (1973): 28-45. Denny, M. "The RelationshipBetween Functional Forms for the ProductionSystern." Canadianjournal of Economics 7 (1974): 21-31. Dielman, Terry E. "Pooled Cross-sectional and Time SeriesData: A Survey of Current StatisticalMethodology." American Statistician 37 (1983): 111-22. Diewert, W. E. "The Application of the ShephardDuality Theorem: A Generalized Leontief Production Function." Journal of Political Economy 79 (1971): 481-507. . "Functional Forms for Profit and TansformationFunctions." journal of Economic Theory 6 (1973): 284-316. . "Applications of Duality Theory. " In Frontiers of Quantitative Economics, edited by Michael D. Intriligator and D. A. Kendrick. Amsterdam: North Holland PublishingCompany, 1974. Fuss, Melvin A. "The Demand for Energy in Canadian Manufacturing." Journal of Econometrics 5 (1979): 89-116. ...... ; Daniel McFadden;and Yair Mundlak. "A Survey of Functional Forms in Economic Analysis of Production." In Production Economics: Dual Approach to Theory and Application_ Vol. 1, edited by Melvin A. Fussand Daniel McFadden, pp. 219.68. Amsterdam: North Holland PublishingCompany, 1978. Gallant, A. Ronald. "Seemingly Unrelated Nonlinear Regressions."journal of Econometrics 3 (1975):35.50. . "On the Bias in Flexible Functional Formsand an EssentiallyUnbiased Form: The Fourier Flexible Form." JournalofEconometrlcs 15 (1981): 211-45. â&#x20AC;˘ "Unbiased Determination of ProductionTechnologies."Journal of Econometrics 20 (1982): 285-323.

98 •

JOURNALOF PHILIPPINEDEVELOPMENT

• "The Fourier Flexible Form." American Journal of Agricultural Economics 66 (1984): 204-8. Griliches, Zvi, and Jerry A. Hausman. Errors in Variables in Panel Data•Technical Working Paper No. 37. National Bureau of Economic Research.Cambridge, Massachusetts,1984.

Guilkey, D. K., and C. A. Lovell. "On the Flexibility of the Translog Approximation." International Economic Review 21 (1980): 137-47. , and R. Sickels." A Comparision of the Performance of Three Flexible Functional Forms." Paper presented at the meeting of the American EconomicsAssociation, New York, December1982. Holt, D.; T. M. F. Smith; and P. D. Winter. "Regression Analysis of Data from Complex Surveys." Journal of the Royal Statistical Society A. 143 Part 4 (1980): 474.87 Jorgenson, Dale W., andLaurence J. Lau. "Duality and Differentiability in Prod uction." Journal Of Economic Theory. 9 (1974): 23-42. "Duality of Technology and Economic Behaviour." Review of Economic Studies 41 (1974): 181-200. Kiefer, Nicholas M. "Quadratic Utility, Labor Supply, and Commodity Demand." In Studies In Nonlinear Estimation, edited by Stephen M. Goldfeld and Richard E. Quandt. Cambridge, Massachusetts: Ballinger Publishing Company, 1976. Kish, Leslie, and Martin Richard Frankel. "Inference from Complex Samples (with Discussion)." Journal of the Royal Statistical Society B. 36 (1974): 1-37. Lau, Laurence J. Comments (to Diewert). Frontiers of Quantitative Economics. Vol. 2: 176-199. Edited by Michael D. Intrilligator and D. A. Kendrick. Amsterdam: North Holland Publishing Company, 1974. . "Applications of Profit Functions." In Production Economics: A Dual Approach to Theory and Applications, Vol. 1, edited by Melvin Fuss ana Daniel McFadden, pp. 217.68. Amsterdam: North Holland Publishing Company, 1978. and Pan A• Yotopoulos. "Profit, Supply and Factor Demand Functions." American Journal of Agricultural Economics 54 (1972): 11.18. Quizon, Jaim¢. "Factor Gains and Lossesin Agriculture: An Application of Cost and Profit Functions." Ph.D. dissertation, School of Economics. University of the Philippines, 1980. Sidhu, Surjit S., and Carlos A. Baanante. "Farm-Level Fertilizer Demand for Mexican Wheat Varieties in the •Indian Punjab." American journal of Agricultural Economics 61 (1979): 455-62. Varian, Hal. Microeconomic Analysis. New York: W. W. Norton and Company, 1978. Wallace, T. D., and A. Hussain. "The Use of Error Component Models in Combining Cross-sectionand Time SeriesData." Econometrica 37 (1969): 55-73. White, Halbert. "Using Least Squaresto Approximate Unknown RegressionFunc-

BANTILAN:PRODUCTIONANALYSIS

tions." International Economic Review 21 (1980) : 149-70. Yotopoulos, Pan A.; Laurence J. Lau; and W. L. Lin. "Microeconomic Output Supply and Factor Demand Functions in the Agriculture of the Provinceof Taiwan." American Journal of Agricultural Economics 58 (1976): 333-40. Zeller, Arnold. "An Efficient Method for Estimating Seemingly Unrelated Rag: ressions and Tests for AggregationBias." Journal of American Statistical Association 57 (1962): 348-68.